I am trying to solve the following Fokker-Planck equation:

$$\frac{\partial}{\partial t} P(x,t|x_0) = \left( \frac{dV(x)}{dx} \frac{\partial}{\partial x} + D \frac{\partial^2}{\partial x^2} \right) P(x,t|x_0) $$ Where $V(x)=\beta |x|$ ,the boundary conditions are $ P(x,t|x_0)=0 $ as $x\rightarrow \pm\infty$ and the initial conditions $P(x,t=0|x_0)=\delta(x-x_0)$

So the FP will become:

$$\frac{\partial}{\partial t} P(x,t|x_0) = \left( \beta \frac{\partial}{\partial x} + D \frac{\partial^2}{\partial x^2} \right) P(x,t|x_0) , x\geq 0 $$

$$\frac{\partial}{\partial t} P(x,t|x_0) = \left( - \beta \frac{\partial}{\partial x} + D \frac{\partial^2}{\partial x^2} \right) P(x,t|x_0) , x< 0 $$

A solution for a similar problem has been solved by Marian Smoluchowski here for the same FP for in half space with reflecting boundary conditions by doing the substitution:

$$P(x,t|x_0)=U e^{-\frac{\beta (x-x_0)}{2D}-\frac{\beta^2t}{4D}}$$ Using this substitution reduces the FP equation to a heat equation:

$$\frac{\partial U}{\partial t}=D \frac{\partial^2U}{\partial x^2}$$

But from here I couldn't proceed with the solution, I have tried to insert the known solutions of this equation in terms of the heat equation in half space but I didn't get to the right solutions, I think I have to change the boundary conditions so it fits the new heat equation, but I am still not completely sure how.. so any tips or suggestions for papers to read would be highly appreciated.

  • $\begingroup$ What are the initial and BC's on U? $\endgroup$ Commented Jun 2, 2019 at 18:30


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